Do deep neural networks have an inbuilt Occam's razor?

The remarkable performance of overparameterized deep neural networks (DNNs) must arise from an interplay between network architecture, training algorithms, and structure in the data. To disentangle these three components, we apply a Bayesian picture, based on the functions expressed by a DNN, to supervised learning. The prior over functions is determined by the network, and is varied by exploiting a transition between ordered and chaotic regimes. For Boolean function classification, we approximate the likelihood using the error spectrum of functions on data. When combined with the prior, this accurately predicts the posterior, measured for DNNs trained with stochastic gradient descent. This analysis reveals that structured data, combined with an intrinsic Occam's razor-like inductive bias towards (Kolmogorov) simple functions that is strong enough to counteract the exponential growth of the number of functions with complexity, is a key to the success of DNNs.

Automatic Gradient Descent: Deep Learning without Hyperparameters

The architecture of a deep neural network is defined explicitly in terms of the number of layers, the width of each layer and the general network topology. Existing optimisation frameworks neglect this information in favour of implicit architectural information (e.g. second-order methods) or architecture-agnostic distance functions (e.g. mirror descent). Meanwhile, the most popular optimiser in practice, Adam, is based on heuristics. This paper builds a new framework for deriving optimisation algorithms that explicitly leverage neural architecture. The theory extends mirror descent to non-convex composite objective functions: the idea is to transform a Bregman divergence to account for the non-linear structure of neural architecture. Working through the details for deep fully-connected networks yields automatic gradient descent: a first-order optimiser without any hyperparameters. Automatic gradient descent trains both fully-connected and convolutional networks out-of-the-box and at ImageNet scale. A PyTorch implementation is available at https://github.com/jxbz/agd and also in Appendix B. Overall, the paper supplies a rigorous theoretical foundation for a next-generation of architecture-dependent optimisers that work automatically and without hyperparameters.

The Equilibrium Hypothesis: Rethinking implicit regularization in Deep Neural Networks

Modern Deep Neural Networks (DNNs) exhibit impressive generalization properties on a variety of tasks without explicit regularization, suggesting the existence of hidden regularization effects. Recent work by Baratin et al. (2021) sheds light on an intriguing implicit regularization effect, showing that some layers are much more aligned with data labels than other layers. This suggests that as the network grows in depth and width, an implicit layer selection phenomenon occurs during training. In this work, we provide the first explanation for this alignment hierarchy. We introduce and empirically validate the Equilibrium Hypothesis which states that the layers that achieve some balance between forward and backward information loss are the ones with the highest alignment to data labels. Our experiments demonstrate an excellent match with the theoretical predictions.

Is SGD a Bayesian sampler? Well, almost

Overparameterised deep neural networks (DNNs) are highly expressive and so can, in principle, generate almost any function that fits a training dataset with zero error. The vast majority of these functions will perform poorly on unseen data, and yet in practice DNNs often generalise remarkably well. This success suggests that a trained DNN must have a strong inductive bias towards functions with low generalisation error. Here we empirically investigate this inductive bias by calculating, for a range of architectures and datasets, the probability $P_{SGD}(f\mid S)$ that an overparameterised DNN, trained with stochastic gradient descent (SGD) or one of its variants, converges on a function $f$ consistent with a training set $S$. We also use Gaussian processes to estimate the Bayesian posterior probability $P_B(f\mid S)$ that the DNN expresses $f$ upon random sampling of its parameters, conditioned on $S$. Our main findings are that $P_{SGD}(f\mid S)$ correlates remarkably well with $P_B(f\mid S)$ and that $P_B(f\mid S)$ is strongly biased towards low-error and low complexity functions. These results imply that strong inductive bias in the parameter-function map (which determines $P_B(f\mid S)$), rather than a special property of SGD, is the primary explanation for why DNNs generalise so well in the overparameterised regime. While our results suggest that the Bayesian posterior $P_B(f\mid S)$ is the first order determinant of $P_{SGD}(f\mid S)$, there remain second order differences that are sensitive to hyperparameter tuning. A function probability picture, based on $P_{SGD}(f\mid S)$ and/or $P_B(f\mid S)$, can shed new light on the way that variations in architecture or hyperparameter settings such as batch size, learning rate, and optimiser choice, affect DNN performance.

Neural networks are a priori biased towards Boolean functions with low entropy

Understanding the inductive bias of neural networks is critical to explaining their ability to generalise. Here, for one of the simplest neural networks -- a single-layer perceptron with $n$ input neurons, one output neuron, and no threshold bias term -- we prove that upon random initialisation of weights, the a priori probability $P(t)$ that it represents a Boolean function that classifies $t$ points in $\{0,1\}^n$ as $1$ has a remarkably simple form: $ P(t) = 2^{-n} \,\, {\rm for} \,\, 0\leq t < 2^n$. Since a perceptron can express far fewer Boolean functions with small or large values of $t$ (low "entropy") than with intermediate values of $t$ (high "entropy") there is, on average, a strong intrinsic a-priori bias towards individual functions with low entropy. Furthermore, within a class of functions with fixed $t$, we often observe a further intrinsic bias towards functions of lower complexity. Finally, we prove that, regardless of the distribution of inputs, the bias towards low entropy becomes monotonically stronger upon adding ReLU layers, and empirically show that increasing the variance of the bias term has a similar effect.

Neural networks are $\textit{a priori}$ biased towards Boolean functions with low entropy

Understanding the inductive bias of neural networks is critical to explaining their ability to generalise. Here, for one of the simplest neural networks -- a single-layer perceptron with $n$ input neurons, one output neuron, and no threshold bias term -- we prove that upon random initialisation of weights, the a priori probability $P(t)$ that it represents a Boolean function that classifies $t$ points in $\{0,1\}^n$ as $1$ has a remarkably simple form: $ P(t) = 2^{-n} \,\, {\rm for} \,\, 0\leq t < 2^n$. Since a perceptron can express far fewer Boolean functions with small or large values of $t$ (low "entropy") than with intermediate values of $t$ (high "entropy") there is, on average, a strong intrinsic a-priori bias towards individual functions with low entropy. Furthermore, within a class of functions with fixed $t$, we often observe a further intrinsic bias towards functions of lower complexity. Finally, we prove that, regardless of the distribution of inputs, the bias towards low entropy becomes monotonically stronger upon adding ReLU layers, and empirically show that increasing the variance of the bias term has a similar effect.